Current International Association of Geodesy efforts within regional geoid determination include the comparison of different computation methods in the quest for the “1-cm geoid.” Internal (formal) and external (empirical) approaches to evaluate geoid errors exist, and ideally they should agree. Spherical radial base functions using the spline kernel (SK), least-squares collocation (LSC), and Stokes’s formula are three commonly used methods for regional geoid computation. The three methods have been shown to be theoretically equivalent, as well as to numerically agree on the millimeter level in a closed-loop environment using synthetic noise-free data (Ophaug and Gerlach in J Geod 91:1367–1382, 2017. https://doi.org/10.1007/s00190-017-1030-1). This companion paper extends the closed-loop method comparison using synthetic data, in that we investigate and compare the formal error propagation using the three methods. We use synthetic uncorrelated and correlated noise regimes, both on the 1-mGal (=10-5ms-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=10^{-5}~{\mathrm {m s}}^{-2}$$\end{document}) level, applied to the input data. The estimated formal errors are validated by comparison with empirical errors, as determined from differences of the noisy geoid solutions to the noise-free solutions. We find that the error propagations of the methods are realistic in both uncorrelated and correlated noise regimes, albeit only when subjected to careful tuning, such as spectral band limitation and signal covariance adaptation. For the SKs, different implementations of the L-curve and generalized cross-validation methods did not provide an optimal regularization parameter. Although the obtained values led to a stabilized numerical system, this was not necessarily equivalent to obtaining the best solution. Using a regularization parameter governed by the agreement between formal and empirical error fields provided a solution of similar quality to the other methods. The errors in the uncorrelated regime are on the level of ∼\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim $$\end{document}5 mm and the method agreement within 1 mm, while the errors in the correlated regime are on the level of ∼\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim $$\end{document}10 mm, and the method agreement within 5 mm. Stokes’s formula generally gives the smallest error, closely followed by LSC and the SKs. To this effect, we note that error estimates from integration and estimation techniques must be interpreted differently, because the latter also take the signal characteristics into account. The high level of agreement gives us confidence in the applicability and comparability of formal errors resulting from the three methods. Finally, we present the error characteristics of geoid height differences derived from the three methods and discuss them qualitatively in relation to GNSS leveling. If applied to real data, this would permit identification of spatial scales for which height information is preferably derived by spirit leveling or GNSS leveling.

中文翻译：

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使用球面样条、最小二乘搭配和斯托克斯公式在区域大地水准面计算中的误差传播

当前国际大地测量协会在区域大地水准面确定方面的工作包括比较不同计算方法以寻找“1 厘米大地水准面”。存在评估大地水准面误差的内部（正式）和外部（经验）方法，理想情况下它们应该一致。使用样条核 (SK)、最小二乘搭配 (LSC) 和斯托克斯公式的球面径向基函数是区域大地水准面计算的三种常用方法。这三种方法已被证明在理论上是等效的，并且在闭环环境中使用合成无噪声数据在毫米级数值上一致（Ophaug 和 Gerlach 在 J Geod 91:1367–1382, 2017. https： //doi.org/10.1007/s00190-017-1030-1）。这篇配套论文使用合成数据扩展了闭环方法的比较，因为我们使用三种方法调查和比较形式错误传播。我们在 1-mGal (=10-5ms-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=10^{-5}~{\mathrm {ms}}^{- 2}$$\end{document}) 级别，应用于输入数据。估计的形式误差通过与经验误差的比较来验证，这是根据噪声大地水准面解决方案与无噪声解决方案的差异确定的。我们发现这些方法的误差传播在不相关和相关的噪声范围中都是现实的，尽管只有在进行仔细调整时，例如谱带限制和信号协方差自适应。对于 SK，L 曲线和广义交叉验证方法的不同实现没有提供最佳正则化参数。尽管获得的值导致了稳定的数值系统，但这并不一定等同于获得最佳解。使用由形式和经验误差域之间的一致性控制的正则化参数提供了与其他方法质量相似的解决方案。不相关机制中的错误处于 ~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} 的水平\usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim $$\end{document} 5 毫米，方法一致在 1 毫米以内，而相关制度中的误差为∼\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{ \oddsidemargin}{-69pt} \begin{document}$$\sim $$\end{document}10mm，方法一致在5mm以内。斯托克斯公式通常给出最小的误差，紧随其后的是 LSC 和 SK。为此，我们注意到来自积分和估计技术的误差估计必须有不同的解释，因为后者也考虑了信号特征。高度一致使我们对三种方法产生的形式错误的适用性和可比性充满信心。最后，我们介绍了由三种方法得出的大地水准面高度差的误差特征，并定性地讨论了它们与 GNSS 水准测量的关系。如果应用于实际数据，这将允许识别空间尺度，其高度信息最好通过水平仪或 GNSS 水平仪导出。高度一致使我们对三种方法产生的形式错误的适用性和可比性充满信心。最后，我们介绍了由三种方法得出的大地水准面高度差的误差特征，并定性地讨论了它们与 GNSS 水准测量的关系。如果应用于实际数据，这将允许识别空间尺度，其高度信息最好通过水平仪或 GNSS 水平仪导出。高度一致使我们对三种方法产生的形式错误的适用性和可比性充满信心。最后，我们介绍了由三种方法得出的大地水准面高度差的误差特征，并定性地讨论了它们与 GNSS 水准测量的关系。如果应用于实际数据，这将允许识别空间尺度，其高度信息最好通过水平仪或 GNSS 水平仪导出。